<p>A natural number is chosen at random from the first 100 natural numbers. The probability that \(x + \dfrac{100}{x} > 50\) is</p>
Step-by-Step Solution
Key Concept: Solve the inequality x + 100/x > 50 by recognizing it as a quadratic after multiplying by x (noting x > 0), then count integers in the solution set within [1,100].
<p><strong>Step 1:</strong> Start with x + 100/x > 50 where x ∈ {1, 2, ..., 100}</p><p><strong>Step 2:</strong> Multiply both sides by x (positive, so inequality direction preserved): x² + 100 > 50x</p><p><strong>Step 3:</strong> Rearrange: x² - 50x + 100 > 0</p><p><strong>Step 4:</strong> Find roots using quadratic formula: x = (50 ± √(2500 - 400))/2 = (50 ± √2100)/2 = (50 ± 10√21)/2 = 25 ± 5√21</p><p><strong>Step 5:</strong> Calculate: √21 ≈ 4.583, so 5√21 ≈ 22.91</p><p>Roots are approximately: x₁ ≈ 25 - 22.91 ≈ 2.09 and x₂ ≈ 25 + 22.91 ≈ 47.91</p><p><strong>Step 6:</strong> Since the parabola opens upward, x² - 50x + 100 > 0 when x < 2.09 or x > 47.91</p><p><strong>Step 7:</strong> Among natural numbers 1-100: x ∈ {1, 2} ∪ {48, 49, ..., 100}</p><p><strong>Step 8:</strong> Count: 2 values + (100 - 48 + 1) = 2 + 53 = 55 values</p><p><strong>Step 9:</strong> Probability = 55/100 = 11/20</p><p>∴ Answer: C</p>
Correct Answer: C